Interpretation of a Discrete de Rham method as a Finite Element System
Snorre H. Christiansen, Francesca Rapetti
TL;DR
The paper reinterprets the DDR method as a conforming Finite Element System by embedding its spaces into a computable $L^2$ product framework. It constructs a local harmonic-extension PDE-based machinery to define the spaces $B^k(T)$ and $A^k(T)$, ensuring trace compatibility, exactness of the sequence, and computability of moments. By bridging DDR, VEM, and FEEC under the FES umbrella, it transfers Strang-type consistency and cohomology results to this unified setting, providing a robust analytical lens for error analysis on polytopal meshes. The work yields a practical, computable inner product that supports stable, convergent discretizations across orders and clarifies the algebraic-topological structure underlying these methods.
Abstract
We show that the DDR method can be interpreted as defining a computable consistent discrete $\mathrm{L}^2$ product on a conforming FES defined by PDEs. Without modifying the numerical method itself, this point of view provides an alternative approach to the analysis. The conformity and consistency properties we prove are stronger than those previously shown. We can also recover some of the other results that have been proved about DDR, from those that have already been proved, in principle, in the general context of FES. We also bring VEM, the Virtual Element Method, into the discussion.